If I plot the graph I can say that $0,-\pi-2\pi$ are three solutions and now on the positive side of $x$ how do I know that the graph of $y=2x$ passes over or intersecting the graph of $\sin x$
2026-03-30 09:54:08.1774864448
find the total number of solutions of $|\sin(|x|)|=x+|x|, x$ belongs to $[-2\pi,2\pi]$
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When $x\leqslant0$, $x+|x|=0$ and therefore in $[-2\pi,0]$ the solutions are those that you mentioned.
When $x>0$, $x+|x|=2x$ and you only have to consider the case in which $0<x\leqslant\frac12$ because after that $x+|x|=2x>1>\bigl|\sin(|x|)\bigr|$. o, what remains to be proved is that$$x\in\left(0,\frac12\right]\implies 2x>\sin(x).$$Hint: derive both sides.